A sparse semi-blind source identification method and its application to Raman spectroscopy for explosives detection

Rapid and reliable detection and identification of unknown chemical substances are critical to homeland security. It is challenging to identify chemical components from a wide range of explosives. There are two key steps involved. One is a non-destructive and informative spectroscopic technique for data acquisition. The other is an associated library of reference features along with a computational method for feature matching and meaningful detection within or beyond the library. In this paper, we develop a new iterative method to identify unknown substances from mixture samples of Raman spectroscopy. In the first step, a constrained least squares method decomposes the data into a sum of linear combination of the known components and a non-negative residual. In the second step, a sparse and convex blind source separation method extracts components geometrically from the residuals. Verification based on the library templates or expert knowledge helps to confirm these components. If necessary, the confirmed meaningful components are fed back into step one to refine the residual and then step two extracts possibly more hidden components. The two steps may be iterated until no more components can be identified. We illustrate the proposed method in processing a set of the so called swept wavelength optical resonant Raman spectroscopy experimental data by a satisfactory blind extraction of a priori unknown chemical explosives from mixture samples. We also test the method on nuclear magnetic resonance (NMR) spectra for chemical compounds identification. © 2013 Published by Elsevier B.V

Global Large Time Self-similarity of a Thermal-Diffusive Combustion System with Critical Nonlinearity

We study the initial value problem of the thermal-diffusive combustion system: $u_{1,t} = u_{1,x,x} - u_1 u^2_2, u_{2,t} = d u_{2,xx} + u_1 u^2_2, x \in R^1$, for non-negative spatially decaying initial data of arbitrary size and for any positive constant $d$. We show that if the initial data decays to zero sufficiently fast at infinity, then the solution $(u_1,u_2)$ converges to a self-similar solution of the reduced system: $u_{1,t} = u_{1,xx} - u_1 u^2_2, u_{2,t} = d u_{2,xx}$, in the large time limit. In particular, $u_1$ decays to zero like ${\cal O}(t^{-\frac{1}{2}-\delta})$, where $\delta > 0$ is an anomalous exponent depending on the initial data, and $u_2$ decays to zero with normal rate ${\cal O}(t^{-\frac{1}{2}})$. The idea of the proof is to combine the a priori estimates for the decay of global solutions with the renormalization group (RG) method for establishing the self-similarity of the solutions in the large time limit.Comment: 22pages, Latex, [email protected],[email protected], [email protected]

The regularity of harmonic maps into spheres and applications to Bernstein problems

We show the regularity of, and derive a-priori estimates for (weakly) harmonic maps from a Riemannian manifold into a Euclidean sphere under the assumption that the image avoids some neighborhood of a half-equator. The proofs combine constructions of strictly convex functions and the regularity theory of quasi-linear elliptic systems. We apply these results to the spherical and Euclidean Bernstein problems for minimal hypersurfaces, obtaining new conditions under which compact minimal hypersurfaces in spheres or complete minimal hypersurfaces in Euclidean spaces are trivial